Annales Henri Poincaré

, Volume 17, Issue 5, pp 1075–1108 | Cite as

Matrix Models from Operators and Topological Strings

  • Marcos MariñoEmail author
  • Szabolcs Zakany


We propose a new family of matrix models whose 1/N expansion captures the all-genus topological string on toric Calabi–Yau threefolds. These matrix models are constructed from the trace class operators appearing in the quantization of the corresponding mirror curves. The fact that they provide a non-perturbative realization of the (standard) topological string follows from a recent conjecture connecting the spectral properties of these operators, to the enumerative invariants of the underlying Calabi–Yau threefolds. We study in detail the resulting matrix models for some simple geometries, like local \({\mathbb{P}^2}\) and local \({\mathbb{F}_2}\), and we verify that their weak ’t Hooft coupling expansion reproduces the topological string free energies near the conifold singularity. These matrix models are formally similar to those appearing in the Fermi-gas formulation of Chern–Simons matter theories, and their 1/N expansion receives non-perturbative corrections determined by the Nekrasov–Shatashvili limit of the refined topological string.


Matrix Model Topological String ABJM Theory Trace Class Operator Hooft Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Dijkgraaf, R., Vafa, C.: Matrix models, topological strings, and supersymmetric gauge theories. Nucl. Phys. B 644, 3 (2002). arXiv:hep-th/0206255
  2. 2.
    Mariño, M.: Chern–Simons theory, matrix integrals, and perturbative three manifold invariants. Commun. Math. Phys. 253, 25 (2004). arXiv:hep-th/0207096
  3. 3.
    Gopakumar, R., Vafa, C.: On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 3, 1415 (1999). arXiv:hep-th/9811131
  4. 4.
    Aganagic, M., Klemm, A., Mariño, M., Vafa, C.: Matrix model as a mirror of Chern–Simons theory. JHEP 0402, 010 (2004). arXiv:hep-th/0211098
  5. 5.
    Grassi, A., Hatsuda, Y., Mariño, M.: Topological Strings from Quantum Mechanics. arXiv:1410.3382 [hep-th]
  6. 6.
    Aganagic, M., Dijkgraaf, R., Klemm, A., Mariño, M., Vafa, C.: Topological strings and integrable hierarchies. Commun. Math. Phys. 261, 451 (2006). arXiv:hep-th/0312085
  7. 7.
    Aganagic, M., Cheng, M.C.N., Dijkgraaf, R., Krefl, D., Vafa, C.: Quantum Geometry of Refined Topological Strings. JHEP 1211, 019 (2012). arXiv:1105.0630 [hep-th]
  8. 8.
    Mironov, A., Morozov, A.: Nekrasov functions and exact Bohr–Zommerfeld integrals. JHEP 1004, 040 (2010). arXiv:0910.5670 [hep-th]
  9. 9.
    Nekrasov, N.A., Shatashvili, S.L.: Quantization of Integrable Systems and Four Dimensional Gauge Theories. arXiv:0908.4052 [hep-th]
  10. 10.
    Kapustin, A., Willett, B., Yaakov, I.: Exact results for Wilson loops in superconformal Chern–Simons theories with matter. JHEP 1003, 089 (2010). arXiv:0909.4559 [hep-th]
  11. 11.
    Aharony, O., Bergman, O., Jafferis, D.L., Maldacena, J.: N = 6 superconformal Chern–Simons-matter theories, M2-branes and their gravity duals. JHEP 0810, 091 (2008). arXiv:0806.1218 [hep-th]
  12. 12.
    Mariño, M.: Lectures on localization and matrix models in supersymmetric Chern–Simons-matter theories. J. Phys. A 44, 463001 (2011). arXiv:1104.0783 [hep-th]
  13. 13.
    Kallen, J., Mariño, M.: Instanton effects and quantum spectral curves. arXiv:1308.6485 [hep-th]
  14. 14.
    Kashaev, R., Mariño, M.: Operators from mirror curves and the quantum dilogarithm. arXiv:1501.01014 [hep-th]
  15. 15.
    Huang, M, Wang, X: Topological strings and quantum spectral problems. JHEP 1409, 150 (2014). arXiv:1406.6178 [hep-th]
  16. 16.
    Kallen, J.: The spectral problem of the ABJ Fermi gas. arXiv:1407.0625 [hep-th]
  17. 17.
    Wang, X., Wang, X., Huang, M.: A Note on Instanton Effects in ABJM Theory. arXiv:1409.4967 [hep-th]
  18. 18.
    Zinn-Justin, J., Jentschura, U.D.: Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions. Ann. Phys. 313, 197 (2004). arXiv:quant-ph/0501136
  19. 19.
    Codesido, S., Grassi, A., Mariño, M.: Exact results in N = 8 Chern–Simons-matter theories and quantum geometry. arXiv:1409.1799 [hep-th]
  20. 20.
    Halmagyi, N., Yasnov, V.: The spectral curve of the lens space matrix model. JHEP 0911, 104 (2009). arXiv:hep-th/0311117
  21. 21.
    Halmagyi, N., Okuda, T., Yasnov, V.: Large N duality, lens spaces and the Chern–Simons matrix model. JHEP 0404, 014 (2004). arXiv:hep-th/0312145
  22. 22.
    Eynard, B.: All orders asymptotic expansion of large partitions. J. Stat. Mech. 0807, P07023 (2008). arXiv:0804.0381 [math-ph]
  23. 23.
    Klemm, A., Sulkowski, P.: Seiberg–Witten theory and matrix models. Nucl. Phys. B 819, 400 (2009). arXiv:0810.4944 [hep-th]
  24. 24.
    Sulkowski, P.: Matrix models for 2* theories. Phys. Rev. D 80, 086006 (2009). arXiv:0904.3064 [hep-th]
  25. 25.
    Eynard, B., Kashani-Poor, A.K., Marchal, O.: A matrix model for the topological string I: deriving the matrix model. Ann. Henri Poincaré 15, 1867 (2014). arXiv:1003.1737 [hep-th]
  26. 26.
    Eynard, B., Kashani-Poor, A.K., Marchal, O.: A matrix model for the topological string II. The spectral curve and mirror geometry. Ann. Henri Poincaré 14, 119 (2013). arXiv:1007.2194 [hep-th]
  27. 27.
    Simon B.: Trace Ideals and Their Applications, 2nd edn. American Mathematical Society, Providence (2000)Google Scholar
  28. 28.
    Zamolodchikov, A.B.: Painlevé III and 2-D polymers. Nucl. Phys. B 432, 427 (1994). arXiv:hep-th/9409108
  29. 29.
    Kostov, I.K.: Solvable statistical models on a random lattice. Nucl. Phys. Proc. Suppl. 45A, 13 (1996). arXiv:hep-th/9509124
  30. 30.
    Grassi, A., Mariño, M.: M-theoretic matrix models. arXiv:1403.4276 [hep-th]
  31. 31.
    Hama, N., Hosomichi, K., Lee, S.: Notes on SUSY gauge theories on three-sphere. JHEP 1103, 127 (2011). arXiv:1012.3512 [hep-th]
  32. 32.
    Jafferis, D.L.: The exact superconformal R-symmetry extremizes Z. JHEP 1205, 159 (2012). arXiv:1012.3210 [hep-th]
  33. 33.
    Huang, M., Klemm, A., Poretschkin, M.: Refined stable pair invariants for E-, M- and [p, q]-strings. JHEP 1311, 112 (2013). arXiv:1308.0619 [hep-th]
  34. 34.
    Huang, M., Klemm, A., Reuter, J., Schiereck, M.: Quantum geometry of del Pezzo surfaces in the Nekrasov–Shatashvili limit. arXiv:1401.4723 [hep-th]
  35. 35.
    Brini, A., Cavalieri, R.: Crepant Resolutions and Open Strings II. arXiv:1407.2571 [math.AG]
  36. 36.
    Faddeev L.D.: Discrete Heisenberg–Weyl group and modular group. Lett. Math. Phys. 34, 249 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Faddeev L.D., Kashaev R.M.: Quantum dilogarithm. Mod. Phys. Lett. A 9, 427 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Kapustin, A., Willett, B., Yaakov, I.: Nonperturbative tests of three-dimensional dualities. JHEP 1010, 013 (2010). arXiv:1003.5694 [hep-th]
  39. 39.
    Mariño, M., Putrov, P.: ABJM theory as a Fermi gas. J. Stat. Mech. 1203, P03001 (2012). arXiv:1110.4066 [hep-th]
  40. 40.
    Andersen, J.E., Kashaev, R.: A TQFT from quantum Teichmüller theory. Commun. Math. Phys. 330, 887 (2014). arXiv:1109.6295 [math.QA]
  41. 41.
    Kostov, I.K.: Exact solution of the six vertex model on a random lattice. Nucl. Phys. B 575, 513 (2000). arXiv:hep-th/9911023
  42. 42.
    Hatsuda, Y., Mariño, M., Moriyama, S., Okuyama, K.: Non-perturbative effects and the refined topological string. JHEP 1409, 168 (2014). arXiv:1306.1734 [hep-th]
  43. 43.
    Hatsuda, Y., Moriyama, S., Okuyama, K.: Instanton effects in ABJM theory from fermi gas approach. JHEP 1301, 158 (2013). arXiv:1211.1251 [hep-th]
  44. 44.
    Kazakov, V.A., Kostov, I.K., Nekrasov, N.A.: D particles, matrix integrals and KP hierarchy. Nucl. Phys. B 557, 413 (1999). arXiv:hep-th/9810035
  45. 45.
    Aganagic, M., Bouchard, V., Klemm, A.: Topological strings and (almost) modular forms. Commun. Math. Phys. 277, 771 (2008). arXiv:hep-th/0607100
  46. 46.
    Ghoshal, D., Vafa, C.: c =  1 string as the topological theory of the conifold. Nucl. Phys. B 453, 121 (1995). arXiv:hep-th/9506122
  47. 47.
    Huang, M, Klemm, A.: Holomorphic anomaly in gauge theories and matrix models. JHEP 0709, 054 (2007). arXiv:hep-th/0605195
  48. 48.
    Hanada, M., Honda, M., Honma, Y., Nishimura, J., Shiba, S., Yoshida, Y.: Numerical studies of the ABJM theory for arbitrary N at arbitrary coupling constant. JHEP 1205, 121 (2012). arXiv:1202.5300 [hep-th]
  49. 49.
    Hatsuda, Y., Okuyama, K.: Probing non-perturbative effects in M-theory. JHEP 1410, 158 (2014). arXiv:1407.3786 [hep-th]
  50. 50.
    Haghighat, B., Klemm, A., Rauch, M.: Integrability of the holomorphic anomaly equations. JHEP 0810, 097 (2008). arXiv:0809.1674 [hep-th]
  51. 51.
    Rodriguez Villegas, F.: Modular Mahler measures, I. In: Topics in Number Theory, pp. 17. Kluwer Academic, Dordrecht (1999)Google Scholar
  52. 52.
    Doran, C., Kerr, M.: Algebraic K-theory of toric hypersurfaces. Commun. Number Theory Phys. 5, 397 (2011). arXiv:0809.4669 [math.AG]
  53. 53.
    Mohri, K., Onjo, Y., Yang, S.K.: Closed submonodromy problems, local mirror symmetry and branes on orbifolds. Rev. Math. Phys. 13, 675 (2001). arXiv:hep-th/0009072
  54. 54.
    Klebanov, I.R., Tseytlin, A.A.: Entropy of near extremal black p-branes. Nucl. Phys. B 475, 164 (1996). arXiv:hep-th/9604089
  55. 55.
    Drukker, N., Mariño, M., Putrov, P.: From weak to strong coupling in ABJM theory. Commun. Math. Phys. 306, 511 (2011). arXiv:1007.3837 [hep-th]
  56. 56.
    Herzog, C.P., Klebanov, I.R., Pufu, S.S., Tesileanu, T.: Multi-matrix models and tri-Sasaki Einstein spaces. Phys. Rev. D 83, 046001 (2011). arXiv:1011.5487 [hep-th]
  57. 57.
    Gu, J., Klemm, A., Mariño, M., Reuter, J.: Exact solutions to quantum spectral curves by topological string theory. (2015). arXiv:1506.09176 [hep-th]
  58. 58.
    Brini, A., Tanzini, A.: Exact results for topological strings on resolved Y**p,q singularities. Commun. Math. Phys. 289, 205 (2009). arXiv:0804.2598 [hep-th]
  59. 59.
    Hatsuda, Y.: Spectral zeta function and non-perturbative effects in ABJM Fermi-gas. arXiv:1503.07883 [hep-th]
  60. 60.
    Kashaev, R., Mariño, M., Zakany, S.: Matrix models from operators and topological strings, 2. arXiv:1505.02243 [hep-th]
  61. 61.
    Mariño, M.: Open string amplitudes and large order behavior in topological string theory. JHEP 0803, 060 (2008). arXiv:hep-th/0612127
  62. 62.
    Bouchard, V., Klemm, A., Mariño, M., Pasquetti, S.: Remodeling the B-model. Commun. Math. Phys. 287, 117 (2009). arXiv:0709.1453 [hep-th]
  63. 63.
    Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. Commun. Number Theor. Phys. 1, 347 (2007). arXiv:math-ph/0702045
  64. 64.
    Drukker, N., Mariño, M., Putrov, P.: Nonperturbative aspects of ABJM theory. JHEP 1111, 141 (2011). arXiv:1103.4844 [hep-th]
  65. 65.
    Couso-Santamarí a, R., Edelstein, J.D., Schiappa, R., Vonk, M.: Resurgent Transseries and the Holomorphic Anomaly. arXiv:1308.1695 [hep-th]
  66. 66.
    Couso-Santamarí a, R., Edelstein, J.D., Schiappa, R., Vonk, M.: Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local \({{\mathbb{C}}{\mathbb{P}}^2}\). arXiv:1407.4821 [hep-th]

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© Springer Basel 2015

Authors and Affiliations

  1. 1.Département de Physique Théorique et Section de MathématiquesUniversité de GenèveGenevaSwitzerland
  2. 2.Département de Physique ThéoriqueUniversité de GenèveGenevaSwitzerland

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